Historically, surfaces were initially defined as subspaces of Euclidean spaces. Often, these surfaces were the locus of zeros of certain functions, usually polynomial functions. Such a definition considered the surface as part of a larger (Euclidean) space, and as such was termed extrinsic.

In the previous section, a surface is defined as a topological space with certain property, namely Hausdorff and locally Euclidean. This topological space is not considered as being a subspace of another space. In this sense, the definition given above, which is the definition that mathematicians use at present, is intrinsic.

A surface defined as intrinsic is not required to satisfy the added constraint of being a subspace of Euclidean space. It seems possible at first glance that there are surfaces defined intrinsically that are not surfaces in the extrinsic sense. However, the Whitney embedding theorem asserts that every surface can in fact be embedded homeomorphically into Euclidean space, in fact into E4. Therefore the extrinsic and intrinsic approaches turn out to be equivalent.//wikipedia

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